Question Number 217030 by ArshadS last updated on 27/Feb/25
$$\mathrm{Find}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{integers}\:\:\mathrm{n}\:\:\mathrm{such}\:\mathrm{that}\:\: \\ $$$$\:\mathrm{n}\:+\:\mathrm{1}\:\:\mathrm{divides}\:\:\mathrm{n}^{\mathrm{2}} \:+\:\mathrm{1} \\ $$
Answered by Ghisom last updated on 27/Feb/25
$$\frac{{n}^{\mathrm{2}} +\mathrm{1}}{{n}+\mathrm{1}}={n}−\mathrm{1}+\frac{\mathrm{2}}{{n}+\mathrm{1}} \\ $$$$\mathrm{for}\:{n}\in\mathbb{Z}\:\mathrm{we}\:\mathrm{get}\:{n}\in\left\{−\mathrm{3},\:−\mathrm{2},\:\mathrm{0},\:\mathrm{1}\right\} \\ $$$$\Rightarrow \\ $$$$\mathrm{answer}\:\mathrm{for}\:{n}>\mathrm{0}\:\mathrm{is}\:{n}=\mathrm{1} \\ $$
Commented by ArshadS last updated on 27/Feb/25
$${nice}!\:{thank}\:{you}\:{sir}! \\ $$